Determining the relative orientation of the horizontal components of seismic sensors is a common problem that limits data analysis and interpretation for several acquisition setups, including linear arrays of geophones deployed in borehole installations or ocean bottom seismometers deployed at the seafloor. To solve this problem we propose a new inversion method based on a complex linear algebra approach. Relative orientation angles are retrieved by minimizing, in a least-squares sense, the l2-norm between the complex traces (hodograms) of adjacent pairs of sensors. This methodology can be applied without restrictions only if the wavefield recorded by each pair of sensors is very similar. In most cases, it is possible to satisfy this condition by low-pass filtering the recorded waveforms. The main advantage of our methodology is that, in the complex domain, the relative orientations of seismic sensors can be viewed as a linear inverse problem, which ensures that the preferred solution corresponds to the global minimum of a misfit function. It is also possible to use simultaneously more than one independent data set (other seismic events) to better constrain the solution of the inverse problem. Furthermore, by a computational point of view, our method results faster than the relative orientation methods based on waveform cross-correlation. After several tests on synthetic data sets we applied successfully our methodology to different types of real data. These applications include the alignment of borehole sensors relative to a Vertical Seismic Profiling (VSP) acquisition and the orientation of Ocean Bottom Seismometers (OBS) relative to a neighbouring land station of known orientation. Using land stations, the absolute orientation of OBS can be retrieved. Finally, as a last application, we checked the correct orientation for land stations of a seismological array in Germany.
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